The general relay channel is among the smallest building blocks of communication networks, yet its capacity is still an open problem. Bounds on the capacity of the general relay channel, and the capacity of some particular classes of relay channels, have been derived in the past. In particular, the expression of the cut-set upper bound, and the generalized block-Markov lower bound were derived for the case of the Frequency-Division Additive White Gaussian Noise (FD-AWGN) relay channel, where the source and the relay transmit in different bands. However, despite a plethora of recent works proposing cooperative strategies for wireless relaying networks and studying their performance in the high Signal-to-Noise Ratio (SNR) regime, the capacity of the multipath fading relay channel remains unknown.
The presently described invention focuses on analyzing the multipath fading relay channel in the non-coherent setting, where neither the source, nor the relay, nor the destination have channel state information (CSI), and in the wideband regime, alternatively named low SNR regime. Indeed, in the wideband regime, power is shared among a large number of degrees of freedom, making the SNR per degree of freedom low. Thus the wideband regime is power limited, but not interference limited on the contrary to the high SNR regime. In the wideband regime, the capacity of the point-to-point AWGN channel and the capacity of the point-to-point non-coherent multipath fading channel were shown to be both equal to the received SNR:
      C    Fading    =            C      AWGN        =                  P                  N          o                    =                        lim                      W            ->            ∞                          ⁢                  W          ⁢                                          ⁢                                    log              ⁡                              (                                  1                  +                                      P                                          WN                      o                                                                      )                                      .                              Moreover, in the wideband limit of fading channels, spread-spectrum signals were shown to achieve poor performance, whereas peaky signals in time and frequency, such as low duty-cycle FSK, along with non-coherent detection, were shown to be capacity optimal. In particular, for the Single Input Multiple Output (SIMO) channel with two receive antennas with respective gains 1 and a2, the capacity is
      C    SIMO    =            (              1        +                  a          2                    )        ⁢                  P                  N          o                    .      Results on multiple user channels in the wideband limit include, the capacity region of the AWGN Broadcast Channel (BC), for which time-sharing was shown to be optimal, and the capacity region of the AWGN Multiple Access Channel (MAC), for which FDMA allows all sources to achieve their point-to-point interference-free capacity to the destination.
Some observations can be drawn from previous works on point-to-point and multiple user channels in the wideband regime: the capacity in the multipath fading case is the same as in the AWGN case, it can be reached in a non-coherent setting, and interference is not an issue.
Coming back to the non-coherent multipath fading relay channel in the wideband regime, two questions naturally arise. The present invention addresses these questions, namely whether the FD-AWGN lower bound can be achieved in the non-coherent multipath fading case and whether the cut-set upper-bound can be reached. Note that in the wideband regime, considering the FD channel is relevant and meets the relay half-duplex constraint. This present invention addresses these questions through three main contributions:
1) A hypergraph model of the wideband multipath fading relay channel is proposed.
2) The hypergraph min-cut is shown to be achieved in the non-coherent wideband multipath fading relay channel by a peaky frequency-binning scheme.
3) The hypergraph min-cut is shown to coincide with the generalized block-Markov lower bound on the capacity of the wideband FD-AWGN relay channel, and in certain channel configurations with the cut-set upper-bound, in which case it is equal to capacity.